Metamath Proof Explorer


Theorem gsumzaddlem

Description: The sum of two group sums. (Contributed by Mario Carneiro, 25-Apr-2016) (Revised by AV, 5-Jun-2019)

Ref Expression
Hypotheses gsumzadd.b 𝐵 = ( Base ‘ 𝐺 )
gsumzadd.0 0 = ( 0g𝐺 )
gsumzadd.p + = ( +g𝐺 )
gsumzadd.z 𝑍 = ( Cntz ‘ 𝐺 )
gsumzadd.g ( 𝜑𝐺 ∈ Mnd )
gsumzadd.a ( 𝜑𝐴𝑉 )
gsumzadd.fn ( 𝜑𝐹 finSupp 0 )
gsumzadd.hn ( 𝜑𝐻 finSupp 0 )
gsumzaddlem.w 𝑊 = ( ( 𝐹𝐻 ) supp 0 )
gsumzaddlem.f ( 𝜑𝐹 : 𝐴𝐵 )
gsumzaddlem.h ( 𝜑𝐻 : 𝐴𝐵 )
gsumzaddlem.1 ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) )
gsumzaddlem.2 ( 𝜑 → ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) )
gsumzaddlem.3 ( 𝜑 → ran ( 𝐹f + 𝐻 ) ⊆ ( 𝑍 ‘ ran ( 𝐹f + 𝐻 ) ) )
gsumzaddlem.4 ( ( 𝜑 ∧ ( 𝑥𝐴𝑘 ∈ ( 𝐴𝑥 ) ) ) → ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) )
Assertion gsumzaddlem ( 𝜑 → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) )

Proof

Step Hyp Ref Expression
1 gsumzadd.b 𝐵 = ( Base ‘ 𝐺 )
2 gsumzadd.0 0 = ( 0g𝐺 )
3 gsumzadd.p + = ( +g𝐺 )
4 gsumzadd.z 𝑍 = ( Cntz ‘ 𝐺 )
5 gsumzadd.g ( 𝜑𝐺 ∈ Mnd )
6 gsumzadd.a ( 𝜑𝐴𝑉 )
7 gsumzadd.fn ( 𝜑𝐹 finSupp 0 )
8 gsumzadd.hn ( 𝜑𝐻 finSupp 0 )
9 gsumzaddlem.w 𝑊 = ( ( 𝐹𝐻 ) supp 0 )
10 gsumzaddlem.f ( 𝜑𝐹 : 𝐴𝐵 )
11 gsumzaddlem.h ( 𝜑𝐻 : 𝐴𝐵 )
12 gsumzaddlem.1 ( 𝜑 → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) )
13 gsumzaddlem.2 ( 𝜑 → ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) )
14 gsumzaddlem.3 ( 𝜑 → ran ( 𝐹f + 𝐻 ) ⊆ ( 𝑍 ‘ ran ( 𝐹f + 𝐻 ) ) )
15 gsumzaddlem.4 ( ( 𝜑 ∧ ( 𝑥𝐴𝑘 ∈ ( 𝐴𝑥 ) ) ) → ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) )
16 1 2 mndidcl ( 𝐺 ∈ Mnd → 0𝐵 )
17 5 16 syl ( 𝜑0𝐵 )
18 1 3 2 mndlid ( ( 𝐺 ∈ Mnd ∧ 0𝐵 ) → ( 0 + 0 ) = 0 )
19 5 17 18 syl2anc ( 𝜑 → ( 0 + 0 ) = 0 )
20 19 adantr ( ( 𝜑𝑊 = ∅ ) → ( 0 + 0 ) = 0 )
21 2 fvexi 0 ∈ V
22 21 a1i ( 𝜑0 ∈ V )
23 11 6 fexd ( 𝜑𝐻 ∈ V )
24 23 suppun ( 𝜑 → ( 𝐹 supp 0 ) ⊆ ( ( 𝐹𝐻 ) supp 0 ) )
25 24 9 sseqtrrdi ( 𝜑 → ( 𝐹 supp 0 ) ⊆ 𝑊 )
26 10 6 22 25 gsumcllem ( ( 𝜑𝑊 = ∅ ) → 𝐹 = ( 𝑥𝐴0 ) )
27 26 oveq2d ( ( 𝜑𝑊 = ∅ ) → ( 𝐺 Σg 𝐹 ) = ( 𝐺 Σg ( 𝑥𝐴0 ) ) )
28 2 gsumz ( ( 𝐺 ∈ Mnd ∧ 𝐴𝑉 ) → ( 𝐺 Σg ( 𝑥𝐴0 ) ) = 0 )
29 5 6 28 syl2anc ( 𝜑 → ( 𝐺 Σg ( 𝑥𝐴0 ) ) = 0 )
30 29 adantr ( ( 𝜑𝑊 = ∅ ) → ( 𝐺 Σg ( 𝑥𝐴0 ) ) = 0 )
31 27 30 eqtrd ( ( 𝜑𝑊 = ∅ ) → ( 𝐺 Σg 𝐹 ) = 0 )
32 10 6 fexd ( 𝜑𝐹 ∈ V )
33 32 suppun ( 𝜑 → ( 𝐻 supp 0 ) ⊆ ( ( 𝐻𝐹 ) supp 0 ) )
34 uncom ( 𝐹𝐻 ) = ( 𝐻𝐹 )
35 34 oveq1i ( ( 𝐹𝐻 ) supp 0 ) = ( ( 𝐻𝐹 ) supp 0 )
36 33 35 sseqtrrdi ( 𝜑 → ( 𝐻 supp 0 ) ⊆ ( ( 𝐹𝐻 ) supp 0 ) )
37 36 9 sseqtrrdi ( 𝜑 → ( 𝐻 supp 0 ) ⊆ 𝑊 )
38 11 6 22 37 gsumcllem ( ( 𝜑𝑊 = ∅ ) → 𝐻 = ( 𝑥𝐴0 ) )
39 38 oveq2d ( ( 𝜑𝑊 = ∅ ) → ( 𝐺 Σg 𝐻 ) = ( 𝐺 Σg ( 𝑥𝐴0 ) ) )
40 39 30 eqtrd ( ( 𝜑𝑊 = ∅ ) → ( 𝐺 Σg 𝐻 ) = 0 )
41 31 40 oveq12d ( ( 𝜑𝑊 = ∅ ) → ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) = ( 0 + 0 ) )
42 6 adantr ( ( 𝜑𝑊 = ∅ ) → 𝐴𝑉 )
43 17 ad2antrr ( ( ( 𝜑𝑊 = ∅ ) ∧ 𝑥𝐴 ) → 0𝐵 )
44 42 43 43 26 38 offval2 ( ( 𝜑𝑊 = ∅ ) → ( 𝐹f + 𝐻 ) = ( 𝑥𝐴 ↦ ( 0 + 0 ) ) )
45 20 mpteq2dv ( ( 𝜑𝑊 = ∅ ) → ( 𝑥𝐴 ↦ ( 0 + 0 ) ) = ( 𝑥𝐴0 ) )
46 44 45 eqtrd ( ( 𝜑𝑊 = ∅ ) → ( 𝐹f + 𝐻 ) = ( 𝑥𝐴0 ) )
47 46 oveq2d ( ( 𝜑𝑊 = ∅ ) → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( 𝐺 Σg ( 𝑥𝐴0 ) ) )
48 47 30 eqtrd ( ( 𝜑𝑊 = ∅ ) → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = 0 )
49 20 41 48 3eqtr4rd ( ( 𝜑𝑊 = ∅ ) → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) )
50 49 ex ( 𝜑 → ( 𝑊 = ∅ → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) )
51 5 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝐺 ∈ Mnd )
52 1 3 mndcl ( ( 𝐺 ∈ Mnd ∧ 𝑧𝐵𝑤𝐵 ) → ( 𝑧 + 𝑤 ) ∈ 𝐵 )
53 52 3expb ( ( 𝐺 ∈ Mnd ∧ ( 𝑧𝐵𝑤𝐵 ) ) → ( 𝑧 + 𝑤 ) ∈ 𝐵 )
54 51 53 sylan ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ ( 𝑧𝐵𝑤𝐵 ) ) → ( 𝑧 + 𝑤 ) ∈ 𝐵 )
55 54 caovclg ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ ( 𝑥𝐵𝑦𝐵 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝐵 )
56 simprl ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ℕ )
57 nnuz ℕ = ( ℤ ‘ 1 )
58 56 57 eleqtrdi ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ ‘ 1 ) )
59 10 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝐹 : 𝐴𝐵 )
60 f1of1 ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝑊 )
61 60 ad2antll ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝑊 )
62 suppssdm ( ( 𝐹𝐻 ) supp 0 ) ⊆ dom ( 𝐹𝐻 )
63 62 a1i ( 𝜑 → ( ( 𝐹𝐻 ) supp 0 ) ⊆ dom ( 𝐹𝐻 ) )
64 9 a1i ( 𝜑𝑊 = ( ( 𝐹𝐻 ) supp 0 ) )
65 dmun dom ( 𝐹𝐻 ) = ( dom 𝐹 ∪ dom 𝐻 )
66 10 fdmd ( 𝜑 → dom 𝐹 = 𝐴 )
67 11 fdmd ( 𝜑 → dom 𝐻 = 𝐴 )
68 66 67 uneq12d ( 𝜑 → ( dom 𝐹 ∪ dom 𝐻 ) = ( 𝐴𝐴 ) )
69 unidm ( 𝐴𝐴 ) = 𝐴
70 68 69 eqtrdi ( 𝜑 → ( dom 𝐹 ∪ dom 𝐻 ) = 𝐴 )
71 65 70 eqtr2id ( 𝜑𝐴 = dom ( 𝐹𝐻 ) )
72 63 64 71 3sstr4d ( 𝜑𝑊𝐴 )
73 72 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝑊𝐴 )
74 f1ss ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝑊𝑊𝐴 ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝐴 )
75 61 73 74 syl2anc ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝐴 )
76 f1f ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝐴𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 )
77 75 76 syl ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 )
78 fco ( ( 𝐹 : 𝐴𝐵𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) → ( 𝐹𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 )
79 59 77 78 syl2anc ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐹𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 )
80 79 ffvelrnda ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹𝑓 ) ‘ 𝑘 ) ∈ 𝐵 )
81 11 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝐻 : 𝐴𝐵 )
82 fco ( ( 𝐻 : 𝐴𝐵𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) → ( 𝐻𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 )
83 81 77 82 syl2anc ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐻𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 )
84 83 ffvelrnda ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻𝑓 ) ‘ 𝑘 ) ∈ 𝐵 )
85 59 ffnd ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝐹 Fn 𝐴 )
86 81 ffnd ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝐻 Fn 𝐴 )
87 6 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 𝐴𝑉 )
88 ovexd ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 1 ... ( ♯ ‘ 𝑊 ) ) ∈ V )
89 inidm ( 𝐴𝐴 ) = 𝐴
90 85 86 77 87 87 88 89 ofco ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) = ( ( 𝐹𝑓 ) ∘f + ( 𝐻𝑓 ) ) )
91 90 fveq1d ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) ‘ 𝑘 ) = ( ( ( 𝐹𝑓 ) ∘f + ( 𝐻𝑓 ) ) ‘ 𝑘 ) )
92 91 adantr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) ‘ 𝑘 ) = ( ( ( 𝐹𝑓 ) ∘f + ( 𝐻𝑓 ) ) ‘ 𝑘 ) )
93 fnfco ( ( 𝐹 Fn 𝐴𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) → ( 𝐹𝑓 ) Fn ( 1 ... ( ♯ ‘ 𝑊 ) ) )
94 85 77 93 syl2anc ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐹𝑓 ) Fn ( 1 ... ( ♯ ‘ 𝑊 ) ) )
95 fnfco ( ( 𝐻 Fn 𝐴𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ) → ( 𝐻𝑓 ) Fn ( 1 ... ( ♯ ‘ 𝑊 ) ) )
96 86 77 95 syl2anc ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐻𝑓 ) Fn ( 1 ... ( ♯ ‘ 𝑊 ) ) )
97 inidm ( ( 1 ... ( ♯ ‘ 𝑊 ) ) ∩ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) = ( 1 ... ( ♯ ‘ 𝑊 ) )
98 eqidd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹𝑓 ) ‘ 𝑘 ) = ( ( 𝐹𝑓 ) ‘ 𝑘 ) )
99 eqidd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻𝑓 ) ‘ 𝑘 ) = ( ( 𝐻𝑓 ) ‘ 𝑘 ) )
100 94 96 88 88 97 98 99 ofval ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹𝑓 ) ∘f + ( 𝐻𝑓 ) ) ‘ 𝑘 ) = ( ( ( 𝐹𝑓 ) ‘ 𝑘 ) + ( ( 𝐻𝑓 ) ‘ 𝑘 ) ) )
101 92 100 eqtrd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) ‘ 𝑘 ) = ( ( ( 𝐹𝑓 ) ‘ 𝑘 ) + ( ( 𝐻𝑓 ) ‘ 𝑘 ) ) )
102 5 ad2antrr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝐺 ∈ Mnd )
103 elfzouz ( 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → 𝑛 ∈ ( ℤ ‘ 1 ) )
104 103 adantl ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑛 ∈ ( ℤ ‘ 1 ) )
105 elfzouz2 ( 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ𝑛 ) )
106 105 adantl ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ♯ ‘ 𝑊 ) ∈ ( ℤ𝑛 ) )
107 fzss2 ( ( ♯ ‘ 𝑊 ) ∈ ( ℤ𝑛 ) → ( 1 ... 𝑛 ) ⊆ ( 1 ... ( ♯ ‘ 𝑊 ) ) )
108 106 107 syl ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 1 ... 𝑛 ) ⊆ ( 1 ... ( ♯ ‘ 𝑊 ) ) )
109 108 sselda ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑛 ) ) → 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) )
110 80 adantlr ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹𝑓 ) ‘ 𝑘 ) ∈ 𝐵 )
111 109 110 syldan ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑛 ) ) → ( ( 𝐹𝑓 ) ‘ 𝑘 ) ∈ 𝐵 )
112 1 3 mndcl ( ( 𝐺 ∈ Mnd ∧ 𝑘𝐵𝑥𝐵 ) → ( 𝑘 + 𝑥 ) ∈ 𝐵 )
113 112 3expb ( ( 𝐺 ∈ Mnd ∧ ( 𝑘𝐵𝑥𝐵 ) ) → ( 𝑘 + 𝑥 ) ∈ 𝐵 )
114 102 113 sylan ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ ( 𝑘𝐵𝑥𝐵 ) ) → ( 𝑘 + 𝑥 ) ∈ 𝐵 )
115 104 111 114 seqcl ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐹𝑓 ) ) ‘ 𝑛 ) ∈ 𝐵 )
116 84 adantlr ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻𝑓 ) ‘ 𝑘 ) ∈ 𝐵 )
117 109 116 syldan ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑛 ) ) → ( ( 𝐻𝑓 ) ‘ 𝑘 ) ∈ 𝐵 )
118 104 117 114 seqcl ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ∈ 𝐵 )
119 fzofzp1 ( 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) → ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) )
120 ffvelrn ( ( ( 𝐹𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ 𝐵 )
121 79 119 120 syl2an ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ 𝐵 )
122 ffvelrn ( ( ( 𝐻𝑓 ) : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐵 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ 𝐵 )
123 83 119 122 syl2an ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ 𝐵 )
124 fvco3 ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) = ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) )
125 77 119 124 syl2an ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) = ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) )
126 fveq2 ( 𝑘 = ( 𝑓 ‘ ( 𝑛 + 1 ) ) → ( 𝐹𝑘 ) = ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) )
127 126 eleq1d ( 𝑘 = ( 𝑓 ‘ ( 𝑛 + 1 ) ) → ( ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ↔ ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) )
128 15 expr ( ( 𝜑𝑥𝐴 ) → ( 𝑘 ∈ ( 𝐴𝑥 ) → ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) ) )
129 128 ralrimiv ( ( 𝜑𝑥𝐴 ) → ∀ 𝑘 ∈ ( 𝐴𝑥 ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) )
130 129 ex ( 𝜑 → ( 𝑥𝐴 → ∀ 𝑘 ∈ ( 𝐴𝑥 ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) ) )
131 130 alrimiv ( 𝜑 → ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑘 ∈ ( 𝐴𝑥 ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) ) )
132 131 ad2antrr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑘 ∈ ( 𝐴𝑥 ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) ) )
133 imassrn ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ ran 𝑓
134 77 adantr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 )
135 134 frnd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ran 𝑓𝐴 )
136 133 135 sstrid ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ 𝐴 )
137 vex 𝑓 ∈ V
138 137 imaex ( 𝑓 “ ( 1 ... 𝑛 ) ) ∈ V
139 sseq1 ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝑥𝐴 ↔ ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ 𝐴 ) )
140 difeq2 ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝐴𝑥 ) = ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) )
141 reseq2 ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝐻𝑥 ) = ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) )
142 141 oveq2d ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝐺 Σg ( 𝐻𝑥 ) ) = ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) )
143 142 sneqd ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → { ( 𝐺 Σg ( 𝐻𝑥 ) ) } = { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } )
144 143 fveq2d ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) = ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) )
145 144 eleq2d ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) ↔ ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) )
146 140 145 raleqbidv ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( ∀ 𝑘 ∈ ( 𝐴𝑥 ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) ↔ ∀ 𝑘 ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) )
147 139 146 imbi12d ( 𝑥 = ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( ( 𝑥𝐴 → ∀ 𝑘 ∈ ( 𝐴𝑥 ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) ) ↔ ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) ) )
148 138 147 spcv ( ∀ 𝑥 ( 𝑥𝐴 → ∀ 𝑘 ∈ ( 𝐴𝑥 ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻𝑥 ) ) } ) ) → ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ⊆ 𝐴 → ∀ 𝑘 ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ) )
149 132 136 148 sylc ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ∀ 𝑘 ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ( 𝐹𝑘 ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) )
150 ffvelrn ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) ⟶ 𝐴 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ 𝐴 )
151 77 119 150 syl2an ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ 𝐴 )
152 fzp1nel ¬ ( 𝑛 + 1 ) ∈ ( 1 ... 𝑛 )
153 75 adantr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝐴 )
154 119 adantl ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) )
155 f1elima ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝐴 ∧ ( 𝑛 + 1 ) ∈ ( 1 ... ( ♯ ‘ 𝑊 ) ) ∧ ( 1 ... 𝑛 ) ⊆ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑓 “ ( 1 ... 𝑛 ) ) ↔ ( 𝑛 + 1 ) ∈ ( 1 ... 𝑛 ) ) )
156 153 154 108 155 syl3anc ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑓 “ ( 1 ... 𝑛 ) ) ↔ ( 𝑛 + 1 ) ∈ ( 1 ... 𝑛 ) ) )
157 152 156 mtbiri ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ¬ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑓 “ ( 1 ... 𝑛 ) ) )
158 151 157 eldifd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ‘ ( 𝑛 + 1 ) ) ∈ ( 𝐴 ∖ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) )
159 127 149 158 rspcdva ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝐹 ‘ ( 𝑓 ‘ ( 𝑛 + 1 ) ) ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) )
160 125 159 eqeltrd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) )
161 138 a1i ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 “ ( 1 ... 𝑛 ) ) ∈ V )
162 11 ad2antrr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝐻 : 𝐴𝐵 )
163 162 136 fssresd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) : ( 𝑓 “ ( 1 ... 𝑛 ) ) ⟶ 𝐵 )
164 13 ad2antrr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) )
165 resss ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ 𝐻
166 165 rnssi ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ ran 𝐻
167 4 cntzidss ( ( ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) ∧ ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ ran 𝐻 ) → ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ ( 𝑍 ‘ ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) )
168 164 166 167 sylancl ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ⊆ ( 𝑍 ‘ ran ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) )
169 104 57 eleqtrrdi ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → 𝑛 ∈ ℕ )
170 f1ores ( ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1𝐴 ∧ ( 1 ... 𝑛 ) ⊆ ( 1 ... ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1-onto→ ( 𝑓 “ ( 1 ... 𝑛 ) ) )
171 153 108 170 syl2anc ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1-onto→ ( 𝑓 “ ( 1 ... 𝑛 ) ) )
172 f1of1 ( ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1-onto→ ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1→ ( 𝑓 “ ( 1 ... 𝑛 ) ) )
173 171 172 syl ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 ↾ ( 1 ... 𝑛 ) ) : ( 1 ... 𝑛 ) –1-1→ ( 𝑓 “ ( 1 ... 𝑛 ) ) )
174 suppssdm ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) supp 0 ) ⊆ dom ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) )
175 dmres dom ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) = ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 )
176 175 a1i ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → dom ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) = ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) )
177 174 176 sseqtrid ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) supp 0 ) ⊆ ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) )
178 inss1 ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) ⊆ ( 𝑓 “ ( 1 ... 𝑛 ) )
179 df-ima ( 𝑓 “ ( 1 ... 𝑛 ) ) = ran ( 𝑓 ↾ ( 1 ... 𝑛 ) )
180 179 a1i ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝑓 “ ( 1 ... 𝑛 ) ) = ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) )
181 178 180 sseqtrid ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝑓 “ ( 1 ... 𝑛 ) ) ∩ dom 𝐻 ) ⊆ ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) )
182 177 181 sstrd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) supp 0 ) ⊆ ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) )
183 eqid ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) supp 0 ) = ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) supp 0 )
184 1 2 3 4 102 161 163 168 169 173 182 183 gsumval3 ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) = ( seq 1 ( + , ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ) ‘ 𝑛 ) )
185 179 eqimss2i ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ⊆ ( 𝑓 “ ( 1 ... 𝑛 ) )
186 cores ( ran ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ⊆ ( 𝑓 “ ( 1 ... 𝑛 ) ) → ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) = ( 𝐻 ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) )
187 185 186 ax-mp ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) = ( 𝐻 ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) )
188 resco ( ( 𝐻𝑓 ) ↾ ( 1 ... 𝑛 ) ) = ( 𝐻 ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) )
189 187 188 eqtr4i ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) = ( ( 𝐻𝑓 ) ↾ ( 1 ... 𝑛 ) )
190 189 fveq1i ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ‘ 𝑘 ) = ( ( ( 𝐻𝑓 ) ↾ ( 1 ... 𝑛 ) ) ‘ 𝑘 )
191 fvres ( 𝑘 ∈ ( 1 ... 𝑛 ) → ( ( ( 𝐻𝑓 ) ↾ ( 1 ... 𝑛 ) ) ‘ 𝑘 ) = ( ( 𝐻𝑓 ) ‘ 𝑘 ) )
192 190 191 eqtrid ( 𝑘 ∈ ( 1 ... 𝑛 ) → ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ‘ 𝑘 ) = ( ( 𝐻𝑓 ) ‘ 𝑘 ) )
193 192 adantl ( ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) ∧ 𝑘 ∈ ( 1 ... 𝑛 ) ) → ( ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ‘ 𝑘 ) = ( ( 𝐻𝑓 ) ‘ 𝑘 ) )
194 104 193 seqfveq ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ∘ ( 𝑓 ↾ ( 1 ... 𝑛 ) ) ) ) ‘ 𝑛 ) = ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) )
195 184 194 eqtr2d ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) = ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) )
196 fvex ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ∈ V
197 196 elsn ( ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ∈ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ↔ ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) = ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) )
198 195 197 sylibr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ∈ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } )
199 3 4 cntzi ( ( ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) ∈ ( 𝑍 ‘ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) ∧ ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ∈ { ( 𝐺 Σg ( 𝐻 ↾ ( 𝑓 “ ( 1 ... 𝑛 ) ) ) ) } ) → ( ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) + ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ) = ( ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) ) )
200 160 198 199 syl2anc ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) + ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ) = ( ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) ) )
201 200 eqcomd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) ) = ( ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) + ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ) )
202 1 3 102 115 118 121 123 201 mnd4g ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑛 ∈ ( 1 ..^ ( ♯ ‘ 𝑊 ) ) ) → ( ( ( seq 1 ( + , ( 𝐹𝑓 ) ) ‘ 𝑛 ) + ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) ) + ( ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) + ( ( 𝐻𝑓 ) ‘ ( 𝑛 + 1 ) ) ) ) = ( ( ( seq 1 ( + , ( 𝐹𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐹𝑓 ) ‘ ( 𝑛 + 1 ) ) ) + ( ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ 𝑛 ) + ( ( 𝐻𝑓 ) ‘ ( 𝑛 + 1 ) ) ) ) )
203 55 55 58 80 84 101 202 seqcaopr3 ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( seq 1 ( + , ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) = ( ( seq 1 ( + , ( 𝐹𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) + ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) )
204 54 59 81 87 87 89 off ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐹f + 𝐻 ) : 𝐴𝐵 )
205 14 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ran ( 𝐹f + 𝐻 ) ⊆ ( 𝑍 ‘ ran ( 𝐹f + 𝐻 ) ) )
206 51 113 sylan ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ ( 𝑘𝐵𝑥𝐵 ) ) → ( 𝑘 + 𝑥 ) ∈ 𝐵 )
207 206 59 81 87 87 89 off ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐹f + 𝐻 ) : 𝐴𝐵 )
208 eldifi ( 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) → 𝑥𝐴 )
209 eqidd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥𝐴 ) → ( 𝐹𝑥 ) = ( 𝐹𝑥 ) )
210 eqidd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥𝐴 ) → ( 𝐻𝑥 ) = ( 𝐻𝑥 ) )
211 85 86 87 87 89 209 210 ofval ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥𝐴 ) → ( ( 𝐹f + 𝐻 ) ‘ 𝑥 ) = ( ( 𝐹𝑥 ) + ( 𝐻𝑥 ) ) )
212 208 211 sylan2 ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( ( 𝐹f + 𝐻 ) ‘ 𝑥 ) = ( ( 𝐹𝑥 ) + ( 𝐻𝑥 ) ) )
213 24 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐹 supp 0 ) ⊆ ( ( 𝐹𝐻 ) supp 0 ) )
214 f1ofo ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –onto𝑊 )
215 forn ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –onto𝑊 → ran 𝑓 = 𝑊 )
216 214 215 syl ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 → ran 𝑓 = 𝑊 )
217 216 9 eqtrdi ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 → ran 𝑓 = ( ( 𝐹𝐻 ) supp 0 ) )
218 217 sseq2d ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 → ( ( 𝐹 supp 0 ) ⊆ ran 𝑓 ↔ ( 𝐹 supp 0 ) ⊆ ( ( 𝐹𝐻 ) supp 0 ) ) )
219 218 ad2antll ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( ( 𝐹 supp 0 ) ⊆ ran 𝑓 ↔ ( 𝐹 supp 0 ) ⊆ ( ( 𝐹𝐻 ) supp 0 ) ) )
220 213 219 mpbird ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐹 supp 0 ) ⊆ ran 𝑓 )
221 21 a1i ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → 0 ∈ V )
222 59 220 87 221 suppssr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( 𝐹𝑥 ) = 0 )
223 33 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐻 supp 0 ) ⊆ ( ( 𝐻𝐹 ) supp 0 ) )
224 223 35 sseqtrrdi ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐻 supp 0 ) ⊆ ( ( 𝐹𝐻 ) supp 0 ) )
225 217 sseq2d ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 → ( ( 𝐻 supp 0 ) ⊆ ran 𝑓 ↔ ( 𝐻 supp 0 ) ⊆ ( ( 𝐹𝐻 ) supp 0 ) ) )
226 225 ad2antll ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( ( 𝐻 supp 0 ) ⊆ ran 𝑓 ↔ ( 𝐻 supp 0 ) ⊆ ( ( 𝐹𝐻 ) supp 0 ) ) )
227 224 226 mpbird ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐻 supp 0 ) ⊆ ran 𝑓 )
228 81 227 87 221 suppssr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( 𝐻𝑥 ) = 0 )
229 222 228 oveq12d ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( ( 𝐹𝑥 ) + ( 𝐻𝑥 ) ) = ( 0 + 0 ) )
230 19 ad2antrr ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( 0 + 0 ) = 0 )
231 212 229 230 3eqtrd ( ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) ∧ 𝑥 ∈ ( 𝐴 ∖ ran 𝑓 ) ) → ( ( 𝐹f + 𝐻 ) ‘ 𝑥 ) = 0 )
232 207 231 suppss ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( ( 𝐹f + 𝐻 ) supp 0 ) ⊆ ran 𝑓 )
233 ovex ( 𝐹f + 𝐻 ) ∈ V
234 233 137 coex ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) ∈ V
235 suppimacnv ( ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) ∈ V ∧ 0 ∈ V ) → ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) supp 0 ) = ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) “ ( V ∖ { 0 } ) ) )
236 235 eqcomd ( ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) ∈ V ∧ 0 ∈ V ) → ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) “ ( V ∖ { 0 } ) ) = ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) supp 0 ) )
237 234 21 236 mp2an ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) “ ( V ∖ { 0 } ) ) = ( ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) supp 0 )
238 1 2 3 4 51 87 204 205 56 75 232 237 gsumval3 ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( seq 1 ( + , ( ( 𝐹f + 𝐻 ) ∘ 𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) )
239 12 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ran 𝐹 ⊆ ( 𝑍 ‘ ran 𝐹 ) )
240 eqid ( ( 𝐹𝑓 ) supp 0 ) = ( ( 𝐹𝑓 ) supp 0 )
241 1 2 3 4 51 87 59 239 56 75 220 240 gsumval3 ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐺 Σg 𝐹 ) = ( seq 1 ( + , ( 𝐹𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) )
242 13 adantr ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ran 𝐻 ⊆ ( 𝑍 ‘ ran 𝐻 ) )
243 eqid ( ( 𝐻𝑓 ) supp 0 ) = ( ( 𝐻𝑓 ) supp 0 )
244 1 2 3 4 51 87 81 242 56 75 227 243 gsumval3 ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐺 Σg 𝐻 ) = ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) )
245 241 244 oveq12d ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) = ( ( seq 1 ( + , ( 𝐹𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) + ( seq 1 ( + , ( 𝐻𝑓 ) ) ‘ ( ♯ ‘ 𝑊 ) ) ) )
246 203 238 245 3eqtr4d ( ( 𝜑 ∧ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) )
247 246 expr ( ( 𝜑 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → ( 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) )
248 247 exlimdv ( ( 𝜑 ∧ ( ♯ ‘ 𝑊 ) ∈ ℕ ) → ( ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) )
249 248 expimpd ( 𝜑 → ( ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) ) )
250 7 8 fsuppun ( 𝜑 → ( ( 𝐹𝐻 ) supp 0 ) ∈ Fin )
251 9 250 eqeltrid ( 𝜑𝑊 ∈ Fin )
252 fz1f1o ( 𝑊 ∈ Fin → ( 𝑊 = ∅ ∨ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) )
253 251 252 syl ( 𝜑 → ( 𝑊 = ∅ ∨ ( ( ♯ ‘ 𝑊 ) ∈ ℕ ∧ ∃ 𝑓 𝑓 : ( 1 ... ( ♯ ‘ 𝑊 ) ) –1-1-onto𝑊 ) ) )
254 50 249 253 mpjaod ( 𝜑 → ( 𝐺 Σg ( 𝐹f + 𝐻 ) ) = ( ( 𝐺 Σg 𝐹 ) + ( 𝐺 Σg 𝐻 ) ) )